Copulas for High Dimensions: Models, Estimation, Inference, and Applications by Dong Hwan Oh Department of Economics Duke University Date: Approved: Andrew J. Patton, Supervisor Tim Bollerslev George Tauchen Shakeeb Khan Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Economics in the Graduate School of Duke University 2014 Abstract Copulas for High Dimensions: Models, Estimation, Inference, and Applications by Dong Hwan Oh Department of Economics Duke University Date: Approved: Andrew J. Patton, Supervisor Tim Bollerslev George Tauchen Shakeeb Khan An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Economics in the Graduate School of Duke University 2014 Copyright c 2014 by Dong Hwan Oh All rights reserved except the rights granted by the Creative Commons Attribution-Noncommercial Licence Abstract The dissertation consists of four chapters that concern topics on copulas for high dimensions. Chapter 1 proposes a new general model for high dimension joint distri- butions of asset returns that utilizes high frequency data and copulas. The depen- dence between returns is decomposed into linear and nonlinear components, which enables the use of high frequency data to accurately measure and forecast linear dependence, and the use of a new class of copulas designed to capture nonlinear de- pendence among the resulting linearly uncorrelated residuals. Estimation of the new class of copulas is conducted using a composite likelihood, making the model feasible even for hundreds of variables. A realistic simulation study verifies that multistage estimation with composite likelihood results in small loss in efficiency and large gain in computation speed. Chapter 2, which is co-authored with Professor Andrew Patton, presents new models for the dependence structure, or copula, of economic variables based on a factor structure. The proposed models are particularly attractive for high dimen- sional applications, involving fifty or more variables. This class of models generally lacks a closed-form density, but analytical results for the implied tail dependence can be obtained using extreme value theory, and estimation via a simulation-based method using rank statistics is simple and fast. We study the finite-sample properties of the estimation method for applications involving up to 100 variables, and apply the model to daily returns on all 100 constituents of the S&P 100 index. We find iv significant evidence of tail dependence, heterogeneous dependence, and asymmetric dependence, with dependence being stronger in crashes than in booms. Chapter 3, which is co-authored with Professor Andrew Patton, considers the estimation of the parameters of a copula via a simulated method of moments type approach. This approach is attractive when the likelihood of the copula model is not known in closed form, or when the researcher has a set of dependence measures or other functionals of the copula that are of particular interest. The proposed ap- proach naturally also nests method of moments and generalized method of moments estimators. Drawing on results for simulation based estimation and on recent work in empirical copula process theory, we show the consistency and asymptotic normality of the proposed estimator, and obtain a simple test of over-identifying restrictions as a goodness-of-fit test. The results apply to both iid and time series data. We analyze the finite-sample behavior of these estimators in an extensive simulation study. Chapter 4, which is co-authored with Professor Andrew Patton, proposes a new class of copula-based dynamic models for high dimension conditional distributions, facilitating the estimation of a wide variety of measures of systemic risk. Our pro- posed models draw on successful ideas from the literature on modelling high di- mension covariance matrices and on recent work on models for general time-varying distributions. Our use of copula-based models enable the estimation of the joint model in stages, greatly reducing the computational burden. We use the proposed new models to study a collection of daily credit default swap (CDS) spreads on 100 U.S. firms over the period 2006 to 2012. We find that while the probability of distress for individual firms has greatly reduced since the financial crisis of 2008-09, the joint probability of distress (a measure of systemic risk) is substantially higher now than in the pre-crisis period. v To my parents: Seung Hyeub Oh and Jung Sook Lee vi Contents Abstract iv List of Tables xi List of Figures xiv Acknowledgements xv 1 Modelling High Dimension Distributions with High Frequency Data and Copulas1 1.1 Introduction................................1 1.2 Joint models for covariances and returns................5 1.2.1 A model for uncorrelated standardized residuals........9 1.2.2 Forecasting models for multivariate covariance matrix..... 13 1.3 Estimation methods and model comparisons.............. 16 1.3.1 Estimation using composite likelihood estimation....... 16 1.3.2 Model selection tests with composite likelihood........ 19 1.3.3 Multistage modelling and estimation.............. 22 1.4 Simulation study............................. 27 1.4.1 Finite sample properties of MCLE for jointly symmetric copulas 27 1.4.2 Finite sample properties of multistage estimation....... 29 1.5 Empirical analysis of S&P 100 equity returns.............. 31 1.5.1 In-sample model selection.................... 35 1.5.2 Out-of-sample model selection.................. 37 vii 1.6 Conclusion................................. 40 1.7 Tables and figures............................. 41 2 Modelling Dependence in High Dimensions with Factor Copulas (co-authored with Andrew Patton) 58 2.1 Introduction................................ 58 2.2 Factor copulas............................... 62 2.2.1 Description of a simple factor copula model.......... 63 2.2.2 A multi-factor copula model................... 64 2.2.3 Tail dependence properties of factor copulas.......... 65 2.2.4 Illustration of some factor copulas................ 69 2.2.5 Non-linear factor copula models................. 71 2.3 A Monte Carlo study of SMM estimation of factor copulas...... 72 2.3.1 Description of the model for the conditional joint distribution 73 2.3.2 Simulation-based estimation of copula models......... 74 2.3.3 Finite-sample properties of SMM estimation of factor copulas 75 2.4 High-dimension copula models for S&P 100 returns.......... 80 2.4.1 Results from equidependence copula specifications....... 82 2.4.2 Results from block equidependence copula specifications.... 84 2.4.3 Measuring systemic risk: Marginal Expected Shortfall..... 87 2.5 Conclusion................................. 89 2.6 Tables and figures............................. 90 3 Simulated Method of Moments Estimation for Copula-Based Mul- tivariate Models (co-authored with Andrew Patton) 107 3.1 Introduction................................ 107 3.2 Simulation-based estimation of copula models............. 111 3.2.1 Definition of the SMM estimator................ 111 viii 3.2.2 Consistency of the SMM estimator............... 113 3.2.3 Asymptotic normality of the SMM estimator.......... 117 3.2.4 Consistent estimation of the asymptotic variance....... 119 3.2.5 A test of overidentifying restrictions............... 120 3.2.6 SMM under model mis-specification............... 122 3.3 Simulation study............................. 124 3.4 Application to the dependence between financial firms......... 128 3.5 Conclusion................................. 132 3.6 Sketch of proofs.............................. 132 3.7 Tables and figures............................. 136 4 Time-Varying Systemic Risk: Evidence from a Dynamic Copula Model of CDS Spreads (co-authored with Andrew Patton) 145 4.1 Introduction................................ 145 4.2 A dynamic copula model for high dimensions.............. 148 4.2.1 Factor copulas........................... 149 4.2.2 \GAS" dynamics......................... 151 4.2.3 Other models for dynamic, high dimension copulas...... 156 4.3 Simulation study............................. 157 4.4 Data description and estimation results................. 159 4.4.1 CDS spreads............................ 159 4.4.2 Summary statistics........................ 160 4.4.3 Conditional mean and variance models............. 161 4.4.4 The CDS \Big Bang"....................... 163 4.4.5 Comparing models for the conditional copula.......... 164 4.5 Time-varying systemic risk........................ 166 4.5.1 Joint probability of distress................... 167 ix 4.5.2 Expected proportion in distress................. 169 4.6 Conclusion................................. 171 4.7 Tables and figures............................. 171 A Appendix to Chapter 1 184 A.1 Proofs................................... 184 A.2 Dynamic conditional correlation (DCC) model............. 190 A.3 Hessian matrix for multistage estimations................ 191 B Appendix to Chapter 2 193 B.1 Proofs................................... 193 B.2 Choice of dependence measures for estimation............. 200 B.3 Additional tables............................. 202 C Appendix to Chapter 3 210 C.1 Proofs................................... 210 C.2 Implementation of the SMM estimator................. 223 C.3 Implementation of MLE for factor copulas............... 225 C.4 Additional tables............................. 226 D Appendix to Chapter 4 231 D.1 Proofs..................................